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Question

The total Number of miles driven each day in the United States on all roads by all vehicles since the year 1980 can be modeled by a linear function. in 1990 approximately 5.76 billion vehicle miles were driven daily. In 2005 approximately 8.31 billion vehicle miles were driven daily.

1) Let t be a number of years after 1980 and M be the number of billion vehicle miles driven daily. Write a linear function M(t).

2) use your answer from 1) to estimate how many billion vehicle miles will be driven daily in 2020.

Answer 1

(1)

Number of vehicles = M .

Number of years aftre 1980 = t .

Number of vehicles in the year 1990 is 5.76 billion (t = 10 , M = 5.76 )

Number of vehicles in the year 2005 is 8.31 billion (t = 25 , M = 8.31)

Since the given function is linear function , we make use of slope formula m = (y₂ – y₁) / (x₂ – x₁) .

(x₁ , y₁)  = (t = 10 , M = 5.76 ) , (x₂ , y₂) = (t = 25 , M = 8.31)

m = (8.31 – 5.76) / (25 – 10) ⇒ (2.55/15) = 0.17 .

Slope of the linear function m = 0.17 .

The equation of the function can be found using slope point form of equation .

slope point form of equation : (y - y₁) = m (x - x₁) .

M - 5.76 = 0.17 (t - 10) 

M - 5.76 = 0.17t - 10*0.17

M - 5.76 = 0.17t - 1.7 

Add 5.76 on each side .

M -5.76 +5.76 = 0.17t - 1.7 + 5.76

M = 0.17t + 4.06

Hence the linear function M(t) = 0.17t + 4.06 .

Answer 2

(2)

The linear function M(t) = 0.17t + 4.06 .                    (From (1) )

Number of years aftre 1980 ⇒ t = 40 .    (In the year 2020)

M(40) = 0.17(40) + 4.06 

M(40) = 6.8 + 4.06

M(40) = 10.86 

Hence the number of vehicles in the year 2020 is 10.86 billion .